# Properties

 Label 330.2.m.d Level $330$ Weight $2$ Character orbit 330.m Analytic conductor $2.635$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$330 = 2 \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 330.m (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.63506326670$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} -\zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} -\zeta_{10} q^{6} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} -\zeta_{10}^{3} q^{4} + \zeta_{10} q^{5} -\zeta_{10} q^{6} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{7} -\zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + q^{10} + ( 2 - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{11} - q^{12} + ( 1 - \zeta_{10}^{3} ) q^{13} + ( \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{14} -\zeta_{10}^{3} q^{15} -\zeta_{10} q^{16} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{17} + \zeta_{10}^{3} q^{18} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{19} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{20} + ( -2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{21} + ( 2 - 4 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{22} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{23} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + \zeta_{10}^{2} q^{25} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{26} + \zeta_{10} q^{27} + ( 1 - 3 \zeta_{10} + \zeta_{10}^{2} ) q^{28} + ( -6 + 6 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{29} -\zeta_{10}^{2} q^{30} + ( 6 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{31} - q^{32} + ( -3 + 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{33} + ( -4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{34} + ( 2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{35} + \zeta_{10}^{2} q^{36} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{37} + ( 1 + \zeta_{10}^{2} ) q^{38} + ( -1 - \zeta_{10}^{2} ) q^{39} -\zeta_{10}^{3} q^{40} + ( 7 \zeta_{10} - 5 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{41} + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{42} + ( 4 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{44} - q^{45} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{46} + ( \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{47} + \zeta_{10}^{3} q^{48} + ( 5 - 3 \zeta_{10} + 5 \zeta_{10}^{2} ) q^{49} + \zeta_{10} q^{50} + ( -4 + 4 \zeta_{10} ) q^{51} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{52} + ( -2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{53} + q^{54} + ( 1 + \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{55} + ( -2 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{56} + ( 1 - \zeta_{10}^{3} ) q^{57} + ( 6 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{58} + ( 1 - \zeta_{10} + 13 \zeta_{10}^{3} ) q^{59} -\zeta_{10} q^{60} + ( 2 - 10 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{61} + ( -2 + 2 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{62} + ( -\zeta_{10} + 3 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{63} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{64} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{66} + ( 8 + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{67} + ( -4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{68} + ( 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{69} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{70} + ( 6 + 4 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{71} + \zeta_{10} q^{72} + ( -8 + 8 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{73} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{74} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{75} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{76} + ( -5 - \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{77} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{78} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{2} q^{80} -\zeta_{10}^{3} q^{81} + ( 7 - 5 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{82} + ( -8 - 8 \zeta_{10}^{2} ) q^{83} + ( 1 - \zeta_{10} + 2 \zeta_{10}^{3} ) q^{84} + ( -4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{85} + ( 4 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{86} + ( 6 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{87} + ( -3 + 2 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{88} + ( -10 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{89} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{90} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} ) q^{92} + ( 2 - 8 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{93} + ( 1 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{94} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{95} + \zeta_{10}^{2} q^{96} + ( -2 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{97} + ( 2 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{98} + ( -2 + 4 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 5q^{7} + q^{8} - q^{9} + O(q^{10})$$ $$4q + q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 5q^{7} + q^{8} - q^{9} + 4q^{10} + 9q^{11} - 4q^{12} + 3q^{13} + 5q^{14} - q^{15} - q^{16} - 8q^{17} + q^{18} + 2q^{19} + q^{20} - 10q^{21} + q^{22} + 6q^{23} - q^{24} - q^{25} + 2q^{26} + q^{27} - 12q^{29} + q^{30} + 2q^{31} - 4q^{32} - 4q^{33} + 8q^{34} - q^{36} + 2q^{37} + 3q^{38} - 3q^{39} - q^{40} + 19q^{41} + 12q^{43} - 11q^{44} - 4q^{45} - 6q^{46} + 4q^{47} + q^{48} + 12q^{49} + q^{50} - 12q^{51} - 2q^{52} - 12q^{53} + 4q^{54} + q^{55} - 10q^{56} + 3q^{57} + 12q^{58} + 16q^{59} - q^{60} - 4q^{61} - 12q^{62} - 5q^{63} - q^{64} + 2q^{65} - q^{66} + 12q^{67} - 8q^{68} + 9q^{69} - 5q^{70} + 22q^{71} + q^{72} - 22q^{73} - 2q^{74} + q^{75} + 2q^{76} - 30q^{77} - 2q^{78} - 10q^{79} + q^{80} - q^{81} + 16q^{82} - 24q^{83} + 5q^{84} - 12q^{85} + 8q^{86} + 12q^{87} - 4q^{88} - 34q^{89} - q^{90} - 5q^{91} - 9q^{92} - 2q^{93} + q^{94} - 2q^{95} - q^{96} + 14q^{97} - 2q^{98} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/330\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$211$$ $$221$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i −0.309017 + 0.951057i 0.809017 − 0.587785i
−0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i −2.92705 2.12663i 0.809017 0.587785i 0.309017 0.951057i 1.00000
91.1 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 0.427051 1.31433i −0.309017 0.951057i −0.809017 + 0.587785i 1.00000
181.1 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i −2.92705 + 2.12663i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000
301.1 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i 0.427051 + 1.31433i −0.309017 + 0.951057i −0.809017 0.587785i 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.m.d 4
3.b odd 2 1 990.2.n.a 4
11.c even 5 1 inner 330.2.m.d 4
11.c even 5 1 3630.2.a.bc 2
11.d odd 10 1 3630.2.a.bi 2
33.h odd 10 1 990.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.d 4 1.a even 1 1 trivial
330.2.m.d 4 11.c even 5 1 inner
990.2.n.a 4 3.b odd 2 1
990.2.n.a 4 33.h odd 10 1
3630.2.a.bc 2 11.c even 5 1
3630.2.a.bi 2 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 5 T_{7}^{3} + 10 T_{7}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(330, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$7$ $$25 + 10 T^{2} + 5 T^{3} + T^{4}$$
$11$ $$121 - 99 T + 41 T^{2} - 9 T^{3} + T^{4}$$
$13$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$17$ $$256 + 192 T + 64 T^{2} + 8 T^{3} + T^{4}$$
$19$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$( -9 - 3 T + T^{2} )^{2}$$
$29$ $$1296 + 648 T + 144 T^{2} + 12 T^{3} + T^{4}$$
$31$ $$1936 - 528 T + 64 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4}$$
$41$ $$3481 - 944 T + 186 T^{2} - 19 T^{3} + T^{4}$$
$43$ $$( 4 - 6 T + T^{2} )^{2}$$
$47$ $$1 + T + 6 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$961 + 403 T + 94 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$32761 - 1991 T + 186 T^{2} - 16 T^{3} + T^{4}$$
$61$ $$5776 + 1064 T + 96 T^{2} + 4 T^{3} + T^{4}$$
$67$ $$( -116 - 6 T + T^{2} )^{2}$$
$71$ $$16 + 32 T + 184 T^{2} - 22 T^{3} + T^{4}$$
$73$ $$5776 + 1368 T + 244 T^{2} + 22 T^{3} + T^{4}$$
$79$ $$400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$4096 + 1024 T + 256 T^{2} + 24 T^{3} + T^{4}$$
$89$ $$( 61 + 17 T + T^{2} )^{2}$$
$97$ $$1936 - 1144 T + 276 T^{2} - 14 T^{3} + T^{4}$$